Maximal $L_p$-regularity for $x$-dependent fractional heat equations with Dirichlet conditions

Abstract: We prove optimal regularity results in $L_p$-based function spaces in space and time for a large class of linear parabolic equations with a nonlocal elliptic operator in bounded domains with limited smoothness. Here the nonlocal operator is given by a strongly elliptic and even pseudodifferential operator of order $2a$ ($0<a<1$) with nonsmooth $x$-dependent coefficients. This includes the prominent case of the fractional Laplacian $(-\Delta)^a$, as well as elliptic operators $(-\nabla \cdot A(x)\nabla+b(x))^a$. The proofs are based on general results on maximal $L_p$-regularity and its relation to $\mathcal{R}$-boundedness of the resolvent of the associated (elliptic) operator. Finally, we apply these results to show existence of strong solutions locally in time for a class of nonlinear nonlocal parabolic equations, which include a fractional nonlinear diffusion equation and a fractional porous medium equation after a transformation. The nonlinear results are new for operators on domains with boundary; the linear results are so when $P$ is $x$-dependent nonsymmetric.